Pairs of Consecutive and Near-Consecutive Powerful Numbers: Pell Families, Towers, and Localization

Abstract

Note: the data deposit ZIP (deposit.zip) is temporarily withdrawn and will be provided again soon in a cleaned and usable English version.We study the pairs n, n+1 of powerful numbers, the pairs n−1, n+1 at distance two, and how their numbers grow. For distance one we establish liminf N_1(x)/log x ≥ 1.6822, with a constant computed from Walker's families, list all pairs up to 2^52 from our own enumeration, and show that the families containing a square alone outgrow a local probabilistic model, which nevertheless matches the counts in the computed range. For distance two we count the pairs in a box of kernels and heights as a sum over towers, derive an explicit lower bound at every height, and prove that the number of pairs at distance two up to x differs by O(x^(2/9)) from the number of kernels m with m | U_1(m), T_1(m) even and T_1(m) ≤ x, so that a bound x^θ for these kernels gives the bound x^max(2/9, θ) for the pairs; under a squarefreeness assumption, the case in which Tao's argument for the bound x^(2/5+o(1)) is weakest lies among these kernels. Every computed number is generated from the deposited data or from a cited constant, and every numbered statement is labeled as proved, taken from the literature, computed, measured, heuristic, or open. The data deposit is part of the record of the collected volume, 10.5281/zenodo.23127547; the text is licensed under CC BY 4.0. Part of a series of four linked records: collected volume 10.5281/zenodo.23127547 · Part I 10.5281/zenodo.23127703 · Part III 10.5281/zenodo.23127707. Further observations on the smallest witnesses and the factor searches on three open building blocks will be published as new versions of this record.